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The diffuse-domain, or smoothed boundary, method is an attractive approachfor solving partial differential equations in complex geometries because of itssimplicity and flexibility. In this method the complex geometry is embeddedinto a larger, regular domain. The original PDE is reformulated using asmoothed characteristic function of the complex domain and source terms areintroduced to approximate the boundary conditions. The reformulated equation,which is independent of the dimension and domain geometry, can be solved bystandard numerical methods and the same solver can be used for any domaingeometry. A challenge is making the method higher-order accurate. For Dirichletboundary conditions, which we focus on here, current implementationsdemonstrate a wide range in their accuracy but generally the methods yield atbest first order accuracy in $\epsilon$, the parameter that characterizes thewidth of the region over which the characteristic function is smoothed.Typically, $\epsilon\propto h$, the grid size. Here, we analyze thediffuse-domain PDEs using matched asymptotic expansions and explain theobserved behaviors. Our analysis also identifies simple modifications to thediffuse-domain PDEs that yield higher-order accuracy in $\epsilon$, e.g.,$O(\epsilon^2)$ in the $L^2$ norm and $O(\epsilon^p)$ with $1.5\le p\le 2$ inthe $L^{\infty}$ norm. Our analytic results are confirmed numerically instationary and moving domains where the level set method is used to capture thedynamics of the domain boundary and to construct the smoothed characteristicfunction.

Higher-order accurate diffuse-domain methods for partial differential equations with Dirichlet boundary conditions in complex, evolving geometries